Finite Group Databasebeta

← #10690589

Subgroups — #10690589

Order 768.

928 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#385C211yesmaximal, characteristic
2C2×#5207C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic
4C2×#4528C212yesmaximal, complemented
5#67613C212yesmaximal, complemented
6#9085—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×C4⋊C4C411yescharacteristic
9C22×C3×C4⋊C4C2211yescharacteristic
10C22×C3×C22⋊C4C411yescharacteristic
11C22×C3×C22⋊C4C411yescharacteristic
12C22×#208—21no—
13C3×#385—21no—
14C3×#385—21no—
15C3×#385—21no—
16C3×#385—21no—
17C2×C3×#95C411yescharacteristic
18C2×C3×#95C2211yescharacteristic
19#5207—24no—
20C2×#830—21no—
21C2×#830—21no—
22C2×#830—21no—
23C2×#830C2212yes—
24C2×#830C2212yes—
25#4528—24no—
26C2×#407—31no—
27C2×#385D311yescharacteristic
28#1779—31no—
29C2×#324—32no—
30#1280—32no—
31C25×C3C2×C411yescharacteristic
32C23×C3×C4—22no—
33C23×C3×C4—22no—
34C23×C3×C4—42no—
35C23×C3×C4—42no—
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4C2×C411yescharacteristic
38C23×C3×C4C2×C411yescharacteristic
39C23×C3×C4C2×C411yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—42no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—22no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—42no—
60C2×#208—24no—
61C2×#208—42no—
62C2×#208—42no—
63C22×C3⋊C8—22no—
64C22×C3⋊C8—22no—
65C3×#95—21no—
66C3×#95—21no—
67C3×#95—21no—
68C3×#95—21no—
69#830—24no—
70#830—22no—
71#830—22no—
72#830—22no—
73#830—42no—
74#830—42no—
75#830D414yes—
76C24×C4C2×D311yescharacteristic
77C22×M16—61no—
78C22×C4⋊C4Dic311yescharacteristic
79C22×C4⋊C4C2×D311yescharacteristic
80C22×C22⋊C4Dic311yescharacteristic
81C22×C22⋊C4Dic311yescharacteristic
82#407—64no—
83C2×C22⋊C8—32no—
84C2×C22⋊C8—32no—
85C2×C22⋊C8—61no—
86C2×C22⋊C8—61no—
87C2×C22⋊C8—61no—
88#385—21no—
89#385—21no—
90#385—21no—
91#385—21no—
92C2×#95Dic311yescharacteristic
93C2×#95C2×D311yescharacteristic
94#324—64no—
95C24×C3—22no—
96C24×C3—22no—
97C24×C3—21no—
98C24×C3—21no—
99C24×C3—22no—
100C24×C3—22no—
101C24×C3—41no—
102C24×C3—41no—
103C24×C3C22⋊C411yescharacteristic
104C24×C3C22⋊C411yescharacteristic
105C24×C3C22×C411yescharacteristic
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—48no—
136C22×C3×C4—48no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4C22⋊C411yescharacteristic
144C22×C3×C4C22⋊C411yescharacteristic
145C22×C3×C4M1612yescomplemented
146C22×C3×C4M1612yescomplemented
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4#4112yes—
151C3×C4⋊C4—42no—
152C3×C4⋊C4—42no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—24no—
158C3×C22⋊C4—42no—
159C3×C22⋊C4—42no—
160#208—48no—
161C2×C3⋊C8—24no—
162C2×C3⋊C8—42no—
163C2×C3⋊C8—42no—
164C2×C3⋊C8—42no—
165C2×C3⋊C8—42no—
166C2×C3⋊C8—42no—
167C25C2×Dic311yescharacteristic
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4C2×Dic311yescharacteristic
173C23×C4C2×Dic311yescharacteristic
174C23×C4C2×Dic311yescharacteristic
175C23×C4C2×Dic311yescharacteristic
176C23×C4C2×Dic311yescharacteristic
177C23×C4#6711yescharacteristic
178C23×C4#6711yescharacteristic
179C23×C4#6711yescharacteristic
180C23×C4#6711yescharacteristic
181C23×C4C22×D311yescharacteristic
182C2×M16—64no—
183C2×M16—122no—
184C2×M16—122no—
185C22×C8—62no—
186C22×C8—62no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—42no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—42no—
201C22⋊C8—34no—
202C22⋊C8—64no—
203C22⋊C8—62no—
204C22⋊C8—62no—
205C22⋊C8—62no—
206C22⋊C8—122no—
207C22⋊C8—122no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—24no—
217C23×C3—24no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—44no—
231C23×C3—44no—
232C23×C3—44no—
233C23×C3—44no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3#9912yes—
239C23×C3#10112yes—
240C23×C3C2×C22⋊C411yescharacteristic
241C23×C3C4×D412yes—
242C23×C3C2×M1611yescharacteristic
243C23×C3C2×M1611yescharacteristic
244C23×C3#13112yes—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—24no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—44no—
262C2×C3×C4—44no—
263C2×C3×C4—44no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—82no—
280C2×C3×C4—82no—
281C2×C3×C4#9912yes—
282C2×C3×C4#12011yescharacteristic
283C2×C3×C4#12112yes—
284C2×C3×C4#12311yescharacteristic
285C3⋊C8—44no—
286C3⋊C8—82no—
287C24—22no—
288C24—22no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—22no—
293C24—41no—
294C24—41no—
295C24#21711yescharacteristic
296C24#21711yescharacteristic
297C24C22×Dic311yescharacteristic
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4#20812yescomplemented
336C22×C4#20812yescomplemented
337C22×C4#21711yescharacteristic
338C22×C4#21711yescharacteristic
339C22×C4D3×D412yes—
340C22×C4#23712yes—
341C22×C4C2×#6711yescharacteristic
342C22×C4C2×#6711yescharacteristic
343M16—128no—
344C2×C8—64no—
345C2×C8—122no—
346C2×C8—122no—
347C2×C8—122no—
348C2×C8—122no—
349C2×C8—122no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—48no—
374C22×C3—48no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3#38611yescharacteristic
384C22×C3#40711yescharacteristic
385C22×C3#40811yescharacteristic
386C22×C3C2×#9911yescharacteristic
387C22×C3C2×#9911yescharacteristic
388C22×C3C2×#10111yescharacteristic
389C22×C3#41311yescharacteristic
390C22×C3#43512yes—
391C22×C3#43612yes—
392C3×C4—21no—
393C3×C4—22no—
394C3×C4—22no—
395C3×C4—21no—
396C3×C4—42no—
397C3×C4—41no—
398C3×C4—42no—
399C3×C4—81no—
400C3×C4—81no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—24no—
406C23—24no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23#83412yes—
428C23#83612yes—
429C23C2×#20811yescharacteristic
430C23C2×#20811yescharacteristic
431C23D4×Dic312yes—
432C23#94812yes—
433C23C2×#21711yescharacteristic
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4#83412yes—
471C2×C4#93511yescharacteristic
472C2×C4#93711yescharacteristic
473C2×C4#93912yes—
474C8—124no—
475C8—242no—
476C2×C3—22no—
477C2×C3—22no—
478C2×C3—42no—
479C2×C3—42no—
480C2×C3#146311yescharacteristic
481C2×C3#146511yescharacteristic
482C2×C3#146811yescharacteristic
483C2×C3#147211yescharacteristic
484C2×C3#187011yescharacteristic
485C2×C3#189811yescharacteristic
486C2×C3#190111yescharacteristic
487C22—22no—
488C22—22no—
489C22—22no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—48no—
502C22—48no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22#501112yes—
512C22#501212yes—
513C22#520711yescharacteristic
514C22C2×#83411yescharacteristic
515C22C2×#83411yescharacteristic
516C22#522011yescharacteristic
517C22C2×#83611yescharacteristic
518C22#523411yescharacteristic
519C22#523711yescharacteristic
520C4—21no—
521C4—22no—
522C4—22no—
523C4—21no—
524C4—42no—
525C4—42no—
526C4—41no—
527C4—81no—
528C4—81no—
529C3#908511yescharacteristic, complemented, Sylow
530C2—22no—
531C2—22no—
532C2—42no—
533C2—42no—
534C2#7155211yescharacteristic
535C2#7160211yescharacteristic
536C2#7161111yescharacteristic
537C2#7167011yescharacteristic
538C2#7169111yescharacteristic
539C2#7169811yescharacteristic
540C2#7170211yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#385C2
socleC23×C3C2×C22⋊C4
prime radicalC25×C3C2×C4
cosocleC23×C3×C4C23
hypercenterC2×#385D3
hypocenterC3#9085
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#385

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C2×#4528C2semidirect
5. #67613C2semidirect
145. C22×C3×C4M16semidirect
146. C22×C3×C4M16semidirect
335. C22×C4#208semidirect
336. C22×C4#208semidirect
529. C3#9085semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#3852
C2×#52072
#721262
C2×#45282
#676132
#90853